Preheat the oven to 350℉ and spray a 9x13-inch casserole dish with cooking spray. Set aside.
Place the chicken on a cutting board and half them lengthwise. Using a meat tenderizer, pound them until the chicken flattens out and is roughly the same thickness all around, about ¼ inch thick. Set aside.
To make the breading, place ¼ cup of flour on one plate and set aside. On a different plate, add the remaining flour, bread crumbs, parmesan cheese, salt, and pepper. Using a fork, mix and set them aside.
Add 3 eggs to a bowl or plate with high sides and whisk to combine. Set aside.
To prepare the chicken, dredge each breast in flour. Using a fork or tongs, transfer the chicken into the eggs and be sure that the egg is sticking to all sides. Then, firmly press each piece into the panko breading. Lay the chicken on a cutting board to set aside. Continue this process until all the chicken is coated.
Heat the olive oil in a large cast iron skillet. When the oil is fragrant, add 3-4 chicken breasts to the pan. Cook for 3 minutes, flip, and cook for an additional 2-3 minutes or until the center reaches 165ºF. Continue this process until all chicken has been cooked.
Slice the chicken into ½-inch thick pieces. Set aside.
Bring a large pot of salted water to a boil. Cook the pasta for 9 minutes, or about 1 minute less than al dente, according to the package instructions. Drain the pasta and add it back to the pot.
Heat a large pan over medium heat. Add the 1 tablespoon of butter, olive oil, and garlic and cook for 1-2 minutes.
Add the remaining butter to the pan. Once it has melted, add the flour and whisk together to create a paste. Cook for 1 minute.
Slowly whisk in the milk and heavy cream. It should thicken slightly. Reduce the heat to low. Add in the salt, pepper, nutmeg, and parmesan cheese and stir to combine until the cheese has melted.
Transfer the sauce and chicken to the pasta. Mix to make sure everything is coated in sauce.
Transfer the pasta and chicken onto the baking dish and top with the mozzarella cheese. Bake for 20 minutes.
Cool for 10 minutes before serving.